Field TheoryStarting with the basic notions and results in algebraic extensions, the authors give an exposition of the work of Galois on the solubility of equations by radicals, including Kummer and Artin-Schreier extensions. This is followed by a Chapter on algebras which contains, among other things, norms and traces of algebra elements for their actions on modules, representations and their characters, and derivations in commutative algebras. The last Chapter deals with transcendence and includes Lüroth's theorem, separability and its connections with derivations. |
Contents
Preface | 1 |
Galois Theory | 78 |
Further Field Theory | 236 |
Derivations and separability | 281 |
Bibliography | 291 |
Common terms and phrases
Abelian algebraic closure algebraic extension algebraically disjoint algebraically independent assertion automorphism cardinality q clearly coefficients contains Corollary deduce denote derivation Derk divisor EK F Esep exists exponent F are linearly field of fractions finite extension fixed field follows Gal(E/K Galois extension Galois group group G hence integral irreducible polynomial isomorphism K-algebra K-algebra homomorphism K-automorphism K-basis K-conjugates K-derivation K-independent K-isomorphism K-linear mapping K-module K]tr K₁ K¹/p linear factors linearly disjoint matrix maximal ideal monic multiplication non-zero element nth root polynomial in K[X prime characteristic prime ideal primitive nth root proof is complete Proposition prove purely inseparable quaternion algebra repeated roots ring root of unity separable extension separably algebraic splits into linear splitting field subalgebra subextension subfield subgroup of G subring subset Suppose surjective theorem topological transcendence basis transcendence degree transcendental unique vanishes vector space